8 1. BASIC FACTS

The j-function plays many important roles. Its values parameterize isomorphism

classes of elliptic curves, and often generate abelian extensions of imaginary qua-

dratic fields [BCHIS, Cox, La1, Shi1]. Its coeﬃcients also appear as the graded

dimensions of explicit representations of the Monster group [Bor1, Bor2, BKO,

AKN]. Here we record some fundamental facts concerning the j-function.

Theorem 1.28. The following are true:

(1) The map z → j(z) is a bijection between F = SL2(Z)\H and C.

(2) The function j(z) is a modular function on SL2(Z).

(3) Every modular function on SL2(Z) is a rational function in j(z).

We conclude this section by recalling the valence formula for meromorphic

modular forms on SL2(Z). To state this formula, we recall the fundamental domain

for the action of SL2(Z):

F := −

1

2

≤ (z)

1

2

and |z| 1 ∪ −

1

2

≤ (z) ≤ 0 and |z| = 1 .

Theorem 1.29. If f(z) is a nonzero weight k meromorphic modular form on

SL2(Z), then

k

12

= ord∞(f) +

1

2

ordi(f) +

1

3

ordω(f) +

τ∈F−{i,ω}

ordτ (f).

Corollary 1.30. The following are all true:

(1) The modular function j(z) has a simple pole at infinity and is holomorphic

on H.

(2) The only zero of j(z) in F is a triple zero at z = ω.

(3) We have

j(i) = 1728.

Proof. By Theorem 1.29, it follows that

1 = ord∞(∆) +

1

2

ordi(∆) +

1

3

ordω(∆) +

τ∈F−{i,ω}

ordτ (∆).

Since ∆(z) is a holomorphic form with ord∞(∆) = 1, it follows that ∆(z) is nonva-

nishing on H. Arguing similarly, the fact that E4(ω) = 0 (see Lemma 1.22) implies

that E4(z) has a simple zero at z = ω and is nonvanishing at ∞ and elsewhere in

F. Conclusions (1) and (2) follow easily from Definition 1.26 and Theorem 1.29.

To verify (3), recall from Remark 1.27 that

j(z) − 1728 =

E6(z)2

∆(z)

.

The valence formula implies that E6(i) = 0, and so it must be that j(i) = 1728.